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You asked what's the difference between Modal verbs and Modal auxiliary verbs. I think that these two terms are not mutually exclusive. Modality has to do with necessity and possibility (of various types, epistemic, deontic, volitive, for example). And there are various expressions which can express modality explicitly. Verb phrase (with or without modal auxiliaries), adjectival phrases, sentence adverbs, some constructions ...
(I'd like to understand 'Modality' as such a notional category. My view is simplistic, but is based on, namely Angelika Kratzer's view (hopefully).
This is what I thought. Others may disagree, though !